This lecture explains the final combinatorial step in the proof of the algebraic Montgomery-Yang problem. After the previous reductions, it remains to exclude the case of four cyclic quotient singularities with local fundamental group orders (2, 3, 5, n). The spin^c-refined Donaldson obstruction gives a strong lattice-theoretic constraint on the associated plumbing lattices. Together with the orbifold Bogomolov-Miyaoka-Yau inequality, this reduces the problem to a systematic case-by-case analysis of possible lattice embeddings. We outline how the resulting constraints on the distinguished vector, the total sum vector, and the partial sum vectors lead to contradictions in all cases, thereby proving Theorem 1.4.
